Convergence theorems on multi-dimensional homogeneous quantum walks

Let d be a natural number. A triple \(\left( \mathcal {H}, \left( U^t \right) _{t \in \mathbb {Z}}, E \right) \) is said to be a d-dimensional quantum walk, if the following conditions hold: The measure E in the last item is called a spectral measure.

For the Hilbert space \(\ell _2(\mathbb {Z}^d) \otimes \mathbb {C}^d\), the measure E is defined as follows: For every subset \(\Omega \subset \mathbb {R}^d\), the operator \(E(\Omega )\) is the orthogonal projection from \(\ell _2(\mathbb {Z}^d) \otimes \mathbb {C}^d\) onto \(\ell _2(\Omega \cap \mathbb {Z}^d) \otimes \mathbb {C}^d\). A unitary operator U on \(\ell _2(\mathbb {Z}^d) \otimes \mathbb {C}^d\) defines a triplet \(\left( \mathcal {H}, \left( U^t \right) _{t \in \mathbb {Z}}, E \right) \), which we call a quantum walk in this paper. \(\square \)

Let X be a bounded operator on \(\mathcal {H}\). The operator X is said to be uniform with respect to E, if the mapping \(\mathbb {R}^d \ni \mathbf {k}\mapsto \alpha _\mathbf {k}(X) \in \mathcal {B}(\mathcal {H})\) is continuous with respect to the operator norm on \(\mathcal {B}(\mathcal {H})\).

For \(i \in \{1, \cdots , d\}\), define the partial derivative \(\partial _i (X)\) by the norm limitif it exists.

Let \(\ell _2(\mathbb {Z}^d)\) be the Hilbert space of all the square summable functions on \(\mathbb {Z}^d\). For \(\mathbf {x}\in \mathbb {Z}^d\), denote by \(\delta _\mathbf {x}\) the definition function of \(\{\mathbf {x}\} \subset \mathbb {Z}^d\). Let E be the standard spectral measure on \(\mathbb {R}^d\) defined in Example 2.2. Let j be an element of \(\{1, \cdots , d\}\). Define the unitary operator \(S_j\) by \(\delta _{\mathbf {x}} \mapsto \delta _{\mathbf {x}+ \mathbf {e}_j}\), where \(\{\mathbf {e}_1, \cdots , \mathbf {e}_j, \cdots , \mathbf {e}_d\}\) stands for the standard basis of \(\mathbb {Z}^d\).

By the straightforward calculation, for every \(\mathbf {k}= (k_1, \cdots , k_j, \cdots , k_d) \in \mathbb {R}^d\), we haveSince \(\exp (\mathbf {i}k_j)\) is a continuous function on \(\mathbb {R}^d\), it turns out that \(S_j\) is uniform.

By Eq. (2), for \(i \ne j\), we haveWe also haveWe can also calculate the higher derivatives as above. It follows that the unitary operator \(S_j\) is smooth.

In particular, the bilateral shift S on \(\ell _2(\mathbb {Z})\) is smooth. \(\square \)

Consider the case that \(d = 1\). Let U be the most famous 1-dimensional quantum walk presented in Eq. (1) acting on \(\ell _2(\mathbb {Z}) \otimes \mathbb {C}^2\). Let E be the standard spectral measure on \(\ell _2(\mathbb {Z}) \otimes \mathbb {C}^2\). Since the bilateral shift S on \(\ell _2(\mathbb {Z})\) is smooth, U is smooth with respect to E. \(\square \)

If \(\partial _i (X)\) exists, then \(X (\mathrm {dom}(D_i)) \subset \mathrm {dom}(D_i)\), and \(\partial _i (X)\) is the unique extension of the commutator \([\mathbf {i}D_i, X] = \mathbf {i}(D_i X - X D_i)\) defined on the domain \(\mathrm {dom}(D_i)\). For every \(\mathbf {k}\in \mathbb {R}^d\), \(\partial _i (\alpha _\mathbf {k}(X))\) exists and is equal to \(\alpha _\mathbf {k}( \partial _i (X))\).

The first assertion is well known. See [8, Lemma 2.4] for example. We calculate \(\partial _i (\alpha _\mathbf {k}(X))\) as follows:\(\square \)

Let X be an operator on \(\mathcal {H}\) which is uniform with respect to E. The operator X is said to be smooth with respect to E, if for every sequence \(\{i(m)\}\) of \(\{1, \cdots , d\}\), the higher-order partial derivativesexist and are uniform with respect to E.

Suppose that \(X, Y \in \mathcal {B}(\mathcal {H})\) are smooth with respect to E. Then, \(X^*\) and XY are also smooth and satisfyfor every \(i \in \{1, \cdots , d\}\).

By Eq. (2), we have the following equation of limits in norm topology:Therefore, every partial derivatives of \(X^*\) exist. Since \(\partial _i (X)\) is uniform, its adjoint is also uniform. Repeating this calculation of partial derivatives, every higher derivatives of \(X^*\) also exist and are uniform. This means that \(X^*\) is also smooth.

Since Y is uniform, the equationholds. Combining with Eq. (2) for X and for Y, we haveTherefore, every partial derivatives of XY exist. Since \(\partial _i(X), Y, X, \partial _i(Y)\) are uniform, \(\partial _i(X Y)\) is also uniform. Repeating this calculation of partial derivatives, every higher derivatives of XY also exist and are uniform. This means that XY is also smooth. \(\square \)

Let X be a bounded operator on \(\mathcal {H}\). The operator X is said to be analytic with respect to E, if the mapping \(\mathbf {k}\mapsto \alpha _\mathbf {k}(X)\) can be extended to a holomorphic mapping defined on a neighborhoodof \(\mathbb {R}^d\), where \(\delta \) is some positive number.

A d-dimensional quantum walk \(\left( \mathcal {H}, \left( U^t \right) _{t \in \mathbb {Z}}, E \right) \) is said to be uniform (smooth, analytic), if U is uniform (smooth, analytic) with respect to E.

Let j be in \(\{1, \cdots , d\}\). Let \(S_j\) be the unitary operator on \(\ell _2(\mathbb {Z}^d)\) given by the shift as in Example 2.5. The action \(\alpha _\mathbf {k}(S_j)\) of \(\mathbf {k}= (k_1, \cdots , k_d) \in \mathbb {R}^d\) is given by Eq. (3), \(\alpha _\mathbf {k}(S_j) = \exp (\mathbf {i}k_j) S_j\). This action can be extended to \(\mathbf {k}= (k_1, \cdots , k_d) \in \mathbb {C}^d\) by \(\mathbf {k}\mapsto \exp (\mathbf {i}k_j) S_j\). This map is holomorphic. It follows that \(S_j\) is analytic. In particular, the bilateral shift S on \(\ell (\mathbb {Z})\) is analytic. Therefore, the most famous 1-dimensional quantum walk presented in Eq. (1) is also analytic.

Two d-dimensional quantum walks \((\mathcal {H}_1, (U_1^t)_{t \in \mathbb {Z}}, E_1)\), \((\mathcal {H}_2, (U_2^t)_{t \in \mathbb {Z}}, E_2)\) are said to be similar, if there exists a unitary operator \(V :\mathcal {H}_1 \rightarrow \mathcal {H}_2\) satisfying the following conditions² :

As in Proposition 3.12, if two quantum walks are similar, and if one of them has a limit distribution, then the other walk also has a limit distribution. Then, the limit is the same as that of the other. For the study of limit distributions of a quantum walk U, we can replace it with another walk which is similar to U. For a Hilbert space \(\mathcal {H}\) with a d-dimensional coordinate system E, we can modify the information of position in the following sense.

Let \(X \subset \mathbb {R}^d\) is a discrete subset. Consider the case that for every \(x \in X_1\), a Hilbert space \(\mathcal {H}_x\) is given. Define a Hilbert space \(\mathcal {H}_X\) by \(\oplus _{x \in X} \mathcal {H}_x\). Let \(E_X\) be the spectral measure on \(\mathbb {R}^d\) defined by the following: For \(\Omega \subset \mathbb {R}^d\), \(E_X(\Omega )\) is the orthogonal projection from \(\mathcal {H}_X\) to \(\oplus _{x \in \Omega \cap X} \mathcal {H}_x\).

Let \(f :X \rightarrow \mathbb {R}^n\) be a map satisfying that there exists a constant \(0 < R\) such that for every \(x \in X\), the distance between x and f(x) is at most R. The map f stands for the modification of position. Define Y by the image f(X). For \(y \in Y\), define \(\mathcal {K}_y\) by \(\oplus _{x \in f^{-1}(y)} \mathcal {H}_x\). Define \(\mathcal {K}_Y\) by \(\oplus _{y \in Y} \mathcal {K}_y\). Let \(E_Y\) be the spectral measure on \(\mathbb {R}^d\) defined by the following: For \(\Omega \subset \mathbb {R}^d\), \(E_Y(\Omega )\) is the orthogonal projection from \(\mathcal {K}_Y\) to \(\oplus _{y \in \Omega \cap Y} \mathcal {K}_y\).

Define a unitary V from \(\mathcal {H}_X = \oplus _{x \in X} \mathcal {H}_x\) to \(\mathcal {K}_Y = \oplus _{y \in Y} \mathcal {K}_y = \oplus _{y \in Y} \oplus _{x \in f^{-1}(y)} \mathcal {H}_x\) by the direct sum of \(\mathrm {id}_{\mathcal {H}_x} :\mathcal {H}_x \rightarrow \mathcal {H}_x\). Then, the unitary V is smooth or more strongly analytic with respect to \(E_X\) and \(E_Y\).

Let U be a quantum walk acting on \((\mathcal {H}_X, E_X)\). Then, we also have a walk \(V U V^{-1}\) acting on \((\mathcal {K}_Y, E_Y)\). Since V is analytic (therefore smooth and uniform), the walk \(V U V^{-1}\) is analytic (smooth, or uniform), if and only if U has the same regularity. Proposition 3.12 shows that asymptotic behavior of \(V U V^{-1}\) is the same as U.

We can apply this example to every crystal lattice. Let d be 2 or 3. Assume that \(X \subset \mathbb {R}^{d}\) forms a crystal lattice. More precisely, there exists an additive subgroup \(G \subset \mathbb {R}^d\) which is isomorphic to \(\mathbb {Z}^d\) such that for every \(g \in G\), \(g + X = X\). We can chose some bounded fundamental domain \(\Xi \subset \mathbb {R}^d\) of the additive action of G on \(\mathbb {R}^d\); that is, the family \(\{g + \Xi \}_{g \in G}\) is disjoint and its union is \(\mathbb {R}^d\). The intersection of \(\Xi \) and X stands for the unit of the crystal structure. Choose a point \(y_0\) in \(\Xi \). Define \(Y \subset \mathbb {R}^d\) by \(\{ g + y_0 \ | \ g \in G\}\). Define \(f :X \rightarrow Y\) as follows: For every \(x \in (g + \Xi ) \cap X\), \(f(x) = g + y_0\). This map f stands for the modification of position.

For a quantum walk U on a Hilbert space associated to X, we can consider a quantum walk \(V U V^{-1}\) on a Hilbert space associated with Y which is similar to U. In this way, we reduce the study of quantum walks on arbitrary crystal lattice to those on the integer lattice \((G + y_0) \cong \mathbb {Z}^d\). \(\square \)

Let \(\left( U^t \right) _{t \in \mathbb {Z}}\) be a unitary representation of \(\mathbb {Z}\) on a Hilbert space \(\mathcal {H}\). A two-sided sequence \(\{c_t\}_{t \in \mathbb {Z}}\) of bounded operators on \(\mathcal {H}\) is called a 1-cocycle of \(\left( U^t \right) _{t \in \mathbb {Z}}\), if for every \(s, t \in \mathbb {Z}\), \(c_{s + t} = U^{-t} c_s U^t + c_t\).

If \(\{c_t\}_{t \in \mathbb {Z}}\) is a 1-cocycle of \(\left( U^t \right) _{t \in \mathbb {Z}}\), then for every \(t \ge 1\), \(\Vert c_t \Vert \le t \Vert c_1\Vert \).

For every \(s, t \in \mathbb {Z}\), we haveRepeating this decomposition, we obtain \(\Vert c_t\Vert \le t \Vert c_1\Vert \). \(\square \)

The limit \(\lim _{t \rightarrow \infty } \frac{\Vert c_t\Vert }{t}\) exists and is less than \(\Vert c_1 \Vert \).

By the inequality (3), the sequence \(\{\Vert c_t\Vert \}\) is subadditive. For every subadditive sequence \(\{\gamma _t\}_{t = 1}^\infty \) of real numbers, the sequence \(\left\{ \gamma _t / t \right\} \) converges to its infimum. \(\square \)

The 1-cocycle \(\left\{ c_t = U^{-t} D U^t - D \right\} \) is called the logarithmic derivatives of the quantum walk \(\left( \mathcal {H}, \left( U^t \right) _{t \in \mathbb {Z}}, E \right) \) with respect to the operator \(D = \sum _i w_i D_i\). The operator \(\left\{ c_t / t \right\} \) is called the average of logarithmic derivatives of the quantum walk.

Let \((\mathcal {H}_1, (U_1^t)_{t \in \mathbb {Z}}, E_1)\) and \((\mathcal {H}_2, (U_2^t)_{t \in \mathbb {Z}}, E_2)\) be d-dimensional smooth quantum walks. Let \(D_i^{(1)}\) be the self-adjoint operator \(\int _{\mathbb {R}^r} x_i E_1(d \mathbf {x})\). Let \(D_i^{(2)}\) be the self-adjoint operator \(\int _{\mathbb {R}^r} x_i E_2(d \mathbf {x})\). Let \(V :\mathcal {H}_1 \rightarrow \mathcal {H}_2\) be a smooth unitary operator which gives similarity between \(U_1\) and \(U_2\). For \(i = 1, \cdots , d\), denote by \(\partial _i^{(1)}\) the i-th partial derivative with respect to \(E_1\) and denote by \(\partial _i^{(2)}\) the i-th partial derivative with respect to \(E_2\). Then for \(i = 1, \cdots , d\), the sequenceconverges to 0 in the norm topology.

By Lemma 2.7, the operator \(\frac{U_2^{-t} \partial _i^{(2)} (U_2^t)}{\mathbf {i}t}\) is equal to the closure ofSince V is smooth with respect to \(E_1\) and \(E_2\), by Lemma 2.7 for \(E_1 \oplus E_2\), the operator \(D_i^{(2)} V - V D_i^{(1)}\) is bounded. This means that distance between \(V^{-1} D_i^{(2)} V\) and \(D_i^{(1)}\) is small. If t is large, the above operator is almost equal toThis is equal to the operator \(V \frac{U_1^{-t} \partial _i^{(1)} (U_1^t)}{\mathbf {i}t} V^{-1}\). \(\square \)

To see what the measure \(p_t\) means, let us look at a quantum walk U acting on \(\ell _2(\mathbb {Z}^d) \otimes \mathbb {C}^n\). Take the spectral measure E as in Example 2.2. Express \(U^t \xi \in \ell _2(\mathbb {Z}^d) \otimes \mathbb {C}^n\) by \((\Psi _t(\mathbf {x}))_{\mathbf {x}\in \mathbb {Z}^d}\), where \(\Psi _t(\mathbf {x})\) is a vector in \(\mathbb {C}^n\). By Eq. (6), for every subset \(\Omega \) of \(\mathbb {R}^d\), we haveTherefore, the measure \(p_t\) is a sum of scalar multiple of point masses \(\{\delta _\mathbf {v}\}_{\mathbf {v}\in \mathbb {Z}^d / t}\), and the coefficient of \(\delta _\mathbf {v}\) is \(\Vert \Psi _t(t \mathbf {v})\Vert ^2\). \(\square \)

Let f be a bounded Borel function. The mean of f with respect to the measure \(p_t\) is given by

In the case that f is a definition function of a Borel subset of \(\mathbb {R}^d\), the above equation holds, by the definition of \(p_t\). It follows that for every Borel step function f, the above equation holds. For a general Borel function f, we have only to use a sequence \(f_n\) of Borel step functions which are uniformly close to f. \(\square \)

A vector \(\xi \) in \(\mathcal {H}\) is said to be smooth, if it is in the domain of \(D_{i(m)} D_{i(m -1)} \cdots D_{i(1)}\) for every natural number m and for every sequence \(\{i(1)\), \(\cdots \), \(i(m)\}\) of \(\{1, \cdots , d\}\).

Suppose that the quantum walk \(\left( \mathcal {H}, \left( U^t \right) _{t \in \mathbb {Z}}, E \right) \) is smooth and that the vector \(\xi \) is smooth. Let t be an arbitrary natural number. Then, every polynomial function g on \(\mathbb {R}^d\) is integrable with respect to \(p_t\), and the integral is given by

We may assume that \(g(x_1, x_2 \cdots , x_d)\) is of the form \(x_{i(m)} x_{i(m -1)} \cdots x_{i(1)}\). In this case, we haveSince U and \(\xi \) are smooth with respect to E, \(U^t \xi \) is in the domain of the above self-adjoint operator. Therefore, the right-hand side is well defined. There exists a countable sum \(f(\mathbf {v})\) of scalar multiples of definition functions which is Borel and uniformly close to \(g(\mathbf {v})\). Then, the difference betweenis a bounded operator with small operator norm. Therefore, \(U^t \xi \) is in the domain of \(\int _{\mathbf {x}\in \mathbb {R}^d} f \left( \frac{\mathbf {x}}{t} \right) E(d \mathbf {x})\). For such f, it is easy to show thatSince f is uniformly close to g, the integral \(\int _{\mathbf {v}\in \mathbb {R}^d} f(\mathbf {v}) p_t(d \mathbf {v})\) is close to \(\int _{\mathbf {v}\in \mathbb {R}^d}\) \(g(\mathbf {v})\) \(p_t(d \mathbf {v})\). \(\square \)

Suppose that the quantum walk \(\left( \mathcal {H}, \left( U^t \right) _{t \in \mathbb {Z}}, E \right) \) is smooth and that the initial unit vector \(\xi \) is smooth. There exists a bounded subset K of \(\mathbb {R}^d\) such that \(\lim _{t \rightarrow \infty } p_t(K) = 1\).

Let i be an arbitrary element of \(\{1, \cdots , d\}\). Let m be an arbitrary natural number. Note that the equationholds, by Lemma 3.9.

The triplet \((\mathcal {H}, \left( U^t \right) _{t \in \mathbb {Z}}, D_i)\) is a one-dimensional quantum walk defined in [8, Definition 2.1]. Therefore, we can employ the theory of one-dimensional quantum walks developed in [8]. As proved in [8, Definition 2.27], for every \(m \in \mathbb {N}\),Take and fix a positive number \(L_i\) larger than \(\Vert [D_i, U]\Vert \). The inequalityimplies that \(\lim _t p_t(\mathbb {R}^{i - 1} \times [- L_i, L_i] \times \mathbb {R}^{d - i}) = 1\). We conclude that \(\lim _t p_t([- L_1, L_1] \times \cdots \times [- L_d, L_d]) = 1\). \(\square \)

Suppose that the quantum walk \(\left( \mathcal {H}, \left( U^t \right) _{t \in \mathbb {Z}}, E \right) \) is smooth and that the initial unit vector \(\xi \) is smooth. Then, the following conditions are equivalent: If the above conditions hold, then these limit distributions coincide and their support is compact.

Let K be a compact subset of \(\mathbb {R}^d\) in Theorem 3.10. Let \(\epsilon \) be an arbitrary positive real number. For every polynomial function g on \(\mathbb {R}^d\), there exists a bounded continuous function f on \(\mathbb {R}^d\) such that \(|f(\mathbf {x}) - g(\mathbf {x})| < \epsilon \) for arbitrary \(\mathbf {x}\in K\). For every bounded continuous function f on \(\mathbb {R}^d\), there exists a polynomial function g on \(\mathbb {R}^d\) such that \(|g(\mathbf {x}) - f(\mathbf {x})| < \epsilon \) for arbitrary \(\mathbf {x}\in K\) by the theorem of Stone–Weierstrass. \(\square \)

Let \((\mathcal {H}_1, (U_1^t)_{t \in \mathbb {Z}}, E_1)\), \((\mathcal {H}_2, (U_2^t)_{t \in \mathbb {Z}}, E_2)\) be two d-dimensional smooth quantum walks. Suppose that there exists a smooth unitary operator \(V :\mathcal {H}_1 \rightarrow \mathcal {H}_2\) which intertwines \(U_1\) and \(U_2\). Let \(\xi \) be a unit vector in \(\mathcal {H}_1\) which is smooth with respect to \(E_1\). Let \(p_t^{(1)}\) be the sequence of probability measures on \(\mathbb {R}^d\) given by \((\mathcal {H}_1, (U_1^t)_{t \in \mathbb {Z}}, E_1)\) and \(\xi \). Let \(p_t^{(2)}\) be the sequence of probability measures on \(\mathbb {R}^d\) given by \((\mathcal {H}_2, (U_2^t)_{t \in \mathbb {Z}}, E_2)\) and \(V \xi \). If \(p_t^{(1)}\) converges in law, then \(p_t^{(2)}\) also converges in law. Furthermore, these limit distributions coincide.

The proof is substantially the same as that of [8, Theorem 2.31]. \(\square \)

A quadruple \((\mathcal {H}, \left( U^t \right) _{t \in \mathbb {Z}}, E, \rho )\) is called a d-dimensional homogeneous quantum walk, if the following conditions hold: The rank of \(E \left( [0, 1)^d \right) \) is called the degree of freedom.

Let \(\left( U^t \right) _{t \in \mathbb {Z}}\) be a homogeneous analytic quantum walk acting on \(\ell _2(\mathbb {Z}^d) \otimes \mathbb {C}^n\). Let \(w_1, \cdots , w_d\) be real numbers. Denote by D the operator \(\sum _i w_i D_i\). For the case that the walk is one-dimensional \((d = 1)\), the above two sequences converge in norm, and the limits are analytic operators.

We make use of the Fourier transform \(\widehat{U}(\mathbf {k}), \mathbf {k}\in \mathbb {T}_{2 \pi }^d\) of the walk. The operator D corresponds to the operator \(\widehat{D} = - \mathbf {i}\sum _i w_i \frac{\partial }{\partial k_i}\). Denote by \(\mathbf {w}\) the real vector \((w_i)\). By Lemma 2.7, the average of the logarithmic derivative \(c_t / t = U^{-t} [D, U^t] / t\) corresponds toWe make use of the directional derivative \(\frac{d}{d s} = \sum _i w_i \frac{\partial }{\partial k_i}\) along the curve of the form \(\mathbb {R}\ni s \mapsto \mathbf {k}_0 + s \mathbf {w}\in \mathbb {T}_{2 \pi }^d\). Choose an arbitrary base point \(\mathbf {k}_0 \in \mathbb {T}_{2 \pi }^d\). Let I be a closed interval in \(\mathbb {R}\) whose interior part includes 0. For a while, we study the behavior of \(\widehat{U}\) on the segment \(\mathbf {k}_0 + I \mathbf {w}\subset \mathbb {T}_{2 \pi }^d\). Making the interval I shorter, we may assume that the mapis injective. Denote by \(\widehat{u}(s)\) the unitary \(\widehat{U}(\mathbf {k}_0 + s \mathbf {w})\). Note that the map \(I \ni s \mapsto \widehat{u}(s)\) can be extended to a holomorphic map defined on a complex domain including I. By Eq. (8), The average of the logarithmic derivative \(\widehat{c_t} / t\) is equal toLet us make use of the analytic perturbation theory by T. Kato. The unitary \(\widehat{u}(s)\) can be decomposed as followsby [9, Theorem II.1.8 and II.1.10]. See also [9, Section II.4.6]. Here, \(\widehat{v}\) is an analytic map to invertible matrices, and the eigenvalue functions \(\lambda _1\), \(\cdots \), \(\lambda _n\) are holomorphic functions defined on a complex domain including I.

A direct calculation yields thatThe second term is equal toAs t tends to infinity, the norm increases linearly. The norms of the first and the third terms are bounded. It follows that for the calculation of the average with respect to time, it suffices to see the second term. By Eq. (9), we haveThe convergence is uniform on the closed interval I. Because \(|\lambda _i(s)|^2 = 1\), \(\lambda _i(s)^{- 1} \cdot \dfrac{d \lambda _i}{d s}(s)\) is in \(\mathbf {i}\mathbb {R}\), as t tends to infinity, the average of logarithmic derivatives \(\widehat{c_t}(\mathbf {k}_0) / t\) converges to a skew self-adjoint matrix \(\mathbf {i}\widehat{H}(\mathbf {k}_0)\), at each point \(\mathbf {k}_0 \in \mathbb {T}_{2 \pi }^d\). The operator norm is uniformly bounded by \(\left\| \sum _i w_i \frac{\partial \widehat{U}}{\partial k_i} \right\| \) due to Lemma 3.2. This upper bound is independent of \(\mathbf {k}_0\). Therefore, the convergence of \(\widehat{c_t}(\mathbf {k}_0) / t\) at each point \(\mathbf {k}_0\) yields that in the strong operator topology. Applying the Fourier transform, we obtain the first assertion.

Consider the case that d is one. Then, the convergence on each closed interval in \(\mathbb {T}_{2 \pi }\) is uniform, and the limit is given by a matrix-valued analytic function as in Eq. (11). Since the one-dimensional torus \(\mathbb {T}_{2 \pi }\) is a union of two segments, the average of logarithmic derivative converges in norm and the limit is analytic.

Let us consider general d again. The inverse Fourier transform of \(U^{-t} \exp \left( \mathbf {i}\frac{D}{t} \right) U^t\) \(\exp \left( - \mathbf {i}\frac{D}{t} \right) \) isFor every \(\mathbb {C}^n\)-valued analytic functions \(\widehat{\xi }(\mathbf {k})\) defined on \(\mathbb {T}_{2 \pi }^d\), we calculateas follows. The \(\mathbb {C}^n\)-valued function \(\exp \left( - t^{-1} \sum _i w_i \frac{\partial }{\partial k_i} \right) \widehat{\xi }\) is given byThis is the Taylor expansion of \(\xi (\mathbf {k}- t^{-1} \mathbf {w})\). Therefore, the unitary \(\exp \left( - t^{-1} \sum _i \frac{\partial }{\partial k_i} \right) \) acts on analytic vectors by the translation of \(t^{-1} \mathbf {w}\). Since analytic vectors are dense in the Hilbert space, the unitary \(\exp \left( - t^{-1} \sum _i \frac{\partial }{\partial k_i} \right) \) acts on every vector in \(L^2(\mathbb {T}_{2 \pi }^d) \otimes \mathbb {C}^n\) by the translation of \(t^{-1} \mathbf {w}\). By the same reason, the unitary \(\exp \left( t^{-1} \sum _i \frac{\partial }{\partial k_i} \right) \) means the translation by \(- t^{-1} \mathbf {w}\). It follows that for every \(\widehat{\xi } \in L^2 \left( \mathbb {T}_{2 \pi }^d \right) \otimes \mathbb {C}^n\), the following equation holds:Since the vector \(\widehat{\xi }\) in \(L^2(\mathbb {T}_{2 \pi }^d) \otimes \mathbb {C}^n\) is arbitrary, we obtain the following equation between two operators:We make use of the restriction \(\widehat{u}(s) = \widehat{U}(\mathbf {k}+ s \mathbf {w})\) on the segment \(\mathbf {k}_0 + I \mathbf {w}\subset \mathbb {T}_{2 \pi }^d\) and its diagonal decomposition \(\widehat{v}(s) \cdot \mathrm {diag}(\lambda _i(s))_i \cdot \widehat{v}(s)^{-1}\) as in Eq. (10). Assume that s is in I. Since I is simply connected, there exist real-valued analytic functions \(l_1\), \(l_2\), \(\cdots \), \(l_n\) such that \(\exp ( \mathbf {i}l_i(s)) = \lambda _i(s)\).

Formula (12) is equal to \(\widehat{u}(0)^{-t} \widehat{u}(t^{-1})^t\). If t is large, then the matrix \(\widehat{v}(t^{-1})\) is close to \(\widehat{v}(0)\) in norm. The unitary matrix \(\widehat{u}(0)^{-t} \widehat{u}(t^{-1})^t\) is uniformly close toNote that the logarithms \(l_1(s)\), \(\cdots \), \(l_n(s)\) are differentiable, since \(\lambda _i(s)\) are analytic.

Therefore, we haveBy Eq. (11) and by the definition of the skew self-adjoint matrix \(\mathbf {i}\widehat{H}(\mathbf {k}_0)\), we have the following convergence at each point \(\mathbf {k}_0\):Because the sequence consists of unitary operators, the convergence at each point implies that in the strong operator topology. Applying the Fourier transform, we obtain the second assertion.

In the case that d is one, the convergence is that in the operator norm topology, since the convergence is uniform on each segment included in the torus. \(\square \)

For any d-dimensional homogeneous analytic quantum walk with finite degree of freedom, and for any initial unit vector, the weak limit of probability measures \(\{p_t\}\) defined in Sect. 3.2 exists.

We first assume that \(\xi \) is smooth. Via the inverse Fourier transform, the walk U corresponds to an \((n \times n)\)-matrix \(\widehat{U}\) whose entries are analytic functions on \(\mathbb {T}_{2 \pi }^d\). The initial unit vector \(\xi \) corresponds to a smooth element \(\widehat{\xi } \in L^2 \left( \mathbb {T}_{2 \pi }^d \right) \otimes \mathbb {C}^n\). For every \(i = 1, \cdots , d\), the self-adjoint operator \({\widehat{D}}_i\) is given by \(- \mathbf {i}\frac{\partial }{ \partial k_i}\). Define \({\widehat{D}}\) by \(- \mathbf {i}\sum _i w_i \frac{\partial }{\partial k_i}\). By Eq. (7) in Lemma 3.7, the mean of the function \(\exp (\mathbf {i}(\cdot , \mathbf {w})_{\mathbb {R}^d})\) on \(\mathbb {R}^d\) with respect to \(p_t\) is equal to the following inner product:Let \(\widehat{H}\) be the limit of the averages of logarithmic derivatives of \(\widehat{U}^t\) with respect to the differential operator \(- \mathbf {i}\sum w_i \frac{\partial }{\partial k_i}\). As in the proof of Proposition 4.2, define self-adjoint operators \(\widehat{H_i}\) byand \(\widehat{H}\) by \(\sum _{i=1}^d w_i \widehat{H_i}\). By Proposition 4.2, the operatorconverges to \(\exp \left( \mathbf {i}\widehat{H} \right) \) in the strong operator topology. The vector \(\exp \left( \mathbf {i}t^{-1} \widehat{D} \right) \widehat{\xi }\) is given by \(\left[ \exp \left( \mathbf {i}t^{-1} \widehat{D} \right) \widehat{\xi } \right] (\mathbf {k}) = \widehat{\xi }(\mathbf {k}+ t^{-1} \mathbf {w})\). It is uniformly close to \(\widehat{\xi }(\mathbf {k})\). As t tends to infinity, the above integral converges toWe obtainWe obtain that for every linear combination g of \(\{ \exp (\mathbf {i}(\cdot , \mathbf {w})_{\mathbb {R}^d}) \ |\ \mathbf {w}\in \mathbb {R}^d \}\),By Theorem 3.10, there exists a compact subset K of \(\mathbb {R}^d\) such thatThe linear span \(\mathrm {span}\{ \exp (\mathbf {i}(\cdot , \mathbf {w})_{\mathbb {R}^d}) \ |\ \mathbf {w}\in \mathbb {R}^d \}\) is the space of trigonometric functions. By the theorem of Stone–Weierstrass, the linear span is dense in C(K) with respect to the supremum norm. It follows that for every bounded continuous function f on \(\mathbb {R}^d\),In the special case that \(\xi \) is smooth, we finish the proof.

Let \(\widetilde{\xi }\) be an initial unit vector in \(\ell _2(\mathbb {Z}^d) \otimes \mathbb {C}^n = \ell _2(\mathbb {Z}^d \rightarrow \mathbb {C}^n)\) which is not necessarily smooth. Let \(\widetilde{p_t}\) be the sequence of probability measures defined by \(U^t\) and \(\widetilde{\xi }\). Let \(\epsilon \) be an arbitrary positive number. Choose a smooth unit vector \(\xi \in \ell _2(\mathbb {Z}^d \rightarrow \mathbb {C}^n)\) satisfying that \(\left\| \xi - \widetilde{\xi } \right\| < \epsilon \). For every element \(\eta \) of \(\ell _2(\mathbb {Z}^d \rightarrow \mathbb {C}^n)\), define an \(\ell _1\) function \(|\eta |^2\) on \(\mathbb {Z}^d\) byBy the inequality \(\left\| U^t \xi - U^t \widetilde{\xi } \right\| < \epsilon \), we have \(\left\| |U^t \xi |^2 - \left| U^t \widetilde{\xi } \right| ^2 \right\| _{\ell _1} < 2 \epsilon \). By Eq. (7), for every bounded Borel function f on \(\mathbb {R}^d\), we haveWe also obtainIt follows that for every bounded continuous function f on \(\mathbb {R}^d\),It follows that the sequence of probability measures \(\left\{ \widetilde{p_t} \right\} \) weakly converges. \(\square \)